which is bounded for any function \(\varphi \in \mathfrak {D}(\mathbb {T}^d)\) . Then it follows from the Corollary 2.1 in [1]} that
\(\varrho v\in C([0,T];L^{\frac{2\gamma }{\gamma +1}}_{\text{weak}}(\mathbb {T}^d)).\) 
